I have a CFA model including 22 indicators and 6 factors. The types of these observed variables are ordinal one with 3 categories. I use MPLUS 2.01 to estimate the model, and it fits good. However, when I wanna set up a second-order factor, it can't work well. Part of original programs are presented as follows:

One problem that I see is that your first-order factors are not identified because you have freed the first factor loading of each one but have not set the metric of the factor by then fixing the factor variances to one. Either one loading needs to be set to one for each factor or the factor variance needs to be set to one.

Another problem is the BY statement for the second-order factor. It looks like you are fixing all of the factor loadings which I don't understand. See Example 17.2 in the Mplus User's Guide to see how a second-order factor model is set up.

I learned in SEM class that a sufficiently high loading of a measured variable onto a latent factor is .3 or higher (though researchers should also consider sample size when using this rule of thumb--so other rules of thumb such as .4, or the requirement that the loading be statistically significant, can also be used).

Is it generally thought that loadings of first-order factors onto a higher-order factor also be .3, in order to justify a higher-order model as opposed to using a group of intercorrelated first-order factors in your structural model? If one first-order factor only loads at .1 or .2, while others load a .3 or higher, would you have to remove the low-loading first-order factors, since your data do not justify your hypothesized model?

Or can a higher-order model be justified by theory alone (whether literature states that the trait being measured should have this higher-order structure)? Or can a higher-order model be justified with statistics other than the loadings of first-order factors, such as good fit of the higher-order model as a whole, based on indices such as the RMSEA, CFI, and TLI?